Thanks a million! This has been bugging me all day. Guess I'm back to square one with my original problem though ....Though I did notice that the rows and columns of those two matrices are linearly dependant....hmm, that might be useful....

In order for the two matricies to equate (A^2 = the zero matrix), when we multiply each column of A by A, we get:

[AC1 AC2 .... ACn] (distributive)

since this equals the zero matrix, each ACi = 0

So, since null A is the set of solution vectors to AX=0, which by default always has the trivial solutionh, given that every element of the resulting matrix is 0, it's colums must be contained within null A (its always a subset, since they are all zeros, which is always in null A).

Given that it's an IFF statement, just turn this around...

col A c null A -> A^2 = 0

Given that col A is a subset of null A:

each Ci is an element of null A

if so, each column of ACi = 0

which is the same as

0 = A[C1....Cn]

which is

0 = AA

0 = A^2

QED

(PS...can a non trivial solution exist such that such that col B c null A?, I don't think so???)

In order for the two matricies to equate (A^2 = the zero matrix), when we multiply each column of A by A, we get:

[AC1 AC2 .... ACn] (distributive)

since this equals the zero matrix, each ACi = 0

So, since null A is the set of solution vectors to AX=0, which by default always has the trivial solutionh, given that every element of the resulting matrix is 0, it's colums must be contained within null A (its always a subset, since they are all zeros, which is always in null A).